When I do save as copy on a drawing I am not prompted to save the dependents, so I am not able to recreate your scenario that you seem to be describing.
Anyway, there is a mistake in your code that you should fix, though I don't know whether it will resolve your original problem. Change this line:
bool = Part.SaveAs(fileNamePath, swSaveAsCurrentVersion, (swSaveAsOptions_Copy & swSaveAsOptions_Silent), Nothing, 0, 0)
To this line, where the ampersand is now a plus sign:
bool = Part.SaveAs(fileNamePath, swSaveAsCurrentVersion, (swSaveAsOptions_Copy + swSaveAsOptions_Silent), Nothing, 0, 0)
The Options argument uses bitmasking to determine the options said, and rather than telling it that you want option "3", which copies and saves silent, you are telling it you want option "21", which updates inactive views, saves referenced models, and saves silent.
The Save As dialog box can be reproduced by making a change to the part without saving, then the drawing prompts to save the part.
I made the change of "& "to "+" and it now stays silent and does not try to save the referenced part.
Thanks again for your help.